Curvature-induced electron localization in developable Möbius-like nanostructures

Korte, AP; Heijden, G. H. M. van der

doi:10.1088/0953-8984/21/49/495301

“…39 Similarly, curvature has been found to influence electron localization, where deep potential wells arise in connection with singularities of the bending-energy density. 40 Since stretchability and curvature are inextricably linked, 6 our results suggest that the electron localization of a quantum Möbius band is directly related to its stretchability and aspect ratio. For bands made of stretchable materials, the combined influences of nontrivial Gaussian curvature and mean curvature may yield unprecedented effects.…”

Section: Conclusion and Discussionmentioning

confidence: 78%

Influence of material stretchability on the equilibrium shape of a Möbius band

Kleiman

¹

,

Hinz²,

Takato

³

et al. 2016

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We use a two-dimensional discrete, lattice-based model to show that Möbius bands made with stretchable materials are less likely to crease or tear. This stems from a delocalization of twisting strain that occurs if stretching is allowed. The associated low-energy configurations provide strategic target shapes for the guided assembly of nanometer and micron scale Möbius bands. To predict macroscopic band shapes for a given material, we establish a connection between stretchability and relevant continuum moduli, leading to insight regarding the practical feasibility of synthesizing Möbius bands from materials with continuum parameters that can be measured experimentally or estimated by upscale averaging.

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“…There is, therefore, significant interest in the relationship between geometry (and topology) and transport and optical properties such as electrical conductance and photoluminescence. In [27] it was shown that in thin conducting sheets electrons increasingly localise to the high-curvature regions as the sheet folds, with creases forming channels for electron transport. Our methods presented in this paper may be used to find how thin free-standing sheets fold and where creases form.…”

Section: Discussionmentioning

confidence: 99%

“…Gravesen & Willatzen [17] computed quantum eigenstates of a particle confined to the surface of a developable Möbius strip and compared their results with earlier calculations by Yakubo et al [62]. They found curvature effects in the form of a splitting of the otherwise doubly degenerate ground state wave function (see also [27]). Thus qualitative changes in the physical properties of Möbius strip structures (for instance nanostrips) may be anticipated and it is of physical interest to know the exact shape of a free-standing strip.…”

mentioning

confidence: 94%

Equilibrium Shapes with Stress Localisation for Inextensible Elastic Möbius and Other Strips

Starostin

¹

,

Heijden

²

2014

J Elast

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We formulate the problem of finding equilibrium shapes of a thin inextensible elastic strip, developing further our previous work on the Möbius strip. By using the isometric nature of the deformation we reduce the variational problem to a second-order onedimensional problem posed on the centreline of the strip. We derive Euler-Lagrange equations for this problem in Euler-Poincaré form and formulate boundary-value problems for closed symmetric one-and two-sided strips. Numerical solutions for the Möbius strip show a singular point of stress localisation on the edge of the strip, a generic response of inextensible elastic sheets under torsional strain. By cutting and pasting operations on the Möbius strip solution, followed by parameter continuation, we construct equilibrium solutions for strips with different linking numbers and with multiple points of stress localisation. Solutions reveal how strips fold into planar or self-contacting shapes as the length-to-width ratio of the strip is decreased. Our results may be relevant for curvature effects on physical properties of extremely thin two-dimensional structures as for instance produced in nanostructured origami.

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“…For ease of reference we collect here together all the equations derived (i.e., Eqs. (16), (17), (23), (24), (27), (28)):…”

Section: Full System Of Equationsmentioning

confidence: 99%

The Mechanics of Ribbons and Möbius Bands

Fosdick

¹

,

Fried

²

2016

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We formulate the problem of finding equilibrium shapes of a thin inextensible elastic strip, developing further our previous work on the Möbius strip. By using the isometric nature of the deformation we reduce the variational problem to a second-order onedimensional problem posed on the centreline of the strip. We derive Euler-Lagrange equations for this problem in Euler-Poincaré form and formulate boundary-value problems for closed symmetric one-and two-sided strips. Numerical solutions for the Möbius strip show a singular point of stress localisation on the edge of the strip, a generic response of inextensible elastic sheets under torsional strain. By cutting and pasting operations on the Möbius strip solution, followed by parameter continuation, we construct equilibrium solutions for strips with different linking numbers and with multiple points of stress localisation. Solutions reveal how strips fold into planar or self-contacting shapes as the length-to-width ratio of the strip is decreased. Our results may be relevant for curvature effects on physical properties of extremely thin two-dimensional structures as for instance produced in nanostructured origami.

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Curvature-induced electron localization in developable Möbius-like nanostructures

Cited by 11 publications

References 33 publications

Influence of material stretchability on the equilibrium shape of a Möbius band

Influence of material stretchability on the equilibrium shape of a Möbius band

Equilibrium Shapes with Stress Localisation for Inextensible Elastic Möbius and Other Strips

The Mechanics of Ribbons and Möbius Bands

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